On Multiparameter Generalized Counting Process and its Time-Changed Variants
Manisha Dhillon, Kuldeep Kumar Kataria
TL;DR
This work introduces the multiparameter generalized counting process (multiparameter GCP) indexed by $\mathbb{R}^d_+$ and proves its key structural property: it admits a representation as a weighted sum $\sum_{j=1}^{k} j\mathscr{N}_j(\mathbf{t})$ of independent multiparameter Poisson processes, linking it to additive Lévy processes. The authors then study integrals of the multiparameter GCP over rectangles via Riemann-Liouville fractional integration, deriving moments and a compound-sum representation, and show a Gaussian limit for small rectangles. Time-changed variants are developed using multiparameter stable and inverse stable subordinators, with explicit pgfs, moments, and fractional differential equations governing their distributions; both univariate and multivariate forms are treated. The results extend univariate fractional/counting processes to a rich multiparameter setting, enabling analysis of multidimensional timing and scaling via fractional calculus and providing closed-form distributional descriptions through Mittag-Leffler functions. Overall, the paper contributes a cohesive framework for multiparameter counting dynamics with time changes, connected to additive Lévy processes and fractional operators, and furnishes explicit formulas useful for theoretical and applied investigations.
Abstract
We introduce and study a multiparameter version of the generalized counting process (GCP), where there is a possibility of finitely many arrivals simultaneously. We call it the multiparameter GCP. In a particular case, it is uniquely represented as a weighted sum of independent multiparameter Poisson processes. For a specific case, we establish a relationship between the multiparameter GCP and the sum of independent GCPs. Some of its time-changed variants are studied where the time-changing components used are the multiparameter stable subordinator and the multiparameter inverse stable subordinator. An integral of the multiparameter GCP is defined, and its asymptotic distribution is obtained.